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Mathematics

Solve the inequation and represent the solution set on the number line.

−3+x≤8x3+2≤143+2x,where x∈I.-3 + x \le \dfrac{8x}{3} + 2 \le \dfrac{14}{3} + 2x, \text{where } x ∈ \bold{I}.

Linear Inequations

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Answer

Solving left side:

−3+x≤8x3+2⇒−3−2≤8x3−x⇒−3−2≤8x−3x3⇒−5≤5x3⇒−1≤x3⇒x≥−3-3 + x \le \dfrac{8x}{3} + 2 \\[0.5em] \Rightarrow -3 -2 \le \dfrac{8x}{3} -x \\[0.5em] \Rightarrow -3 -2 \le \dfrac{8x-3x}{3} \\[0.5em] \Rightarrow -5 \le \dfrac{5x}{3} \\[0.5em] \Rightarrow -1 \le \dfrac{x}{3} \\[0.5em] \Rightarrow x \ge -3

Solving right side:

8x3+2≤143+2x⇒8x3−2x≤143−2⇒8x−6x3≤14−63⇒2x3≤83⇒2x≤8⇒x≤4\dfrac{8x}{3} + 2 \le \dfrac{14}{3}+2x \\[0.5em] \Rightarrow \dfrac{8x}{3} - 2x \le \dfrac{14}{3} -2 \\[0.5em] \Rightarrow \dfrac{8x-6x}{3} \le \dfrac{14-6}{3} \\[0.5em] \Rightarrow \dfrac{2x}{3} \le \dfrac{8}{3} \\[0.5em] \Rightarrow 2x \le 8 \\[0.5em] \Rightarrow x \le 4

∴ Solution set = {x : x ∈ I, -3 ≤ x ≤ 4} = {-3, -2, -1, 0, 1, 2, 3, 4}

The graph of the solution set of is represented by thick black dots.

Solve the inequation and represent the solution set on the number line: -3 + x ≤ (8x/3) + 2 ≤ (14/3) + 2x, where x ∈ I. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

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